Mountain Lions are like Sharks of the land. What I mean by that is that their danger is wildly overblown in movies and news.
Statistically speaking, attacks from both animals are insanely rare.
For example, there have only been 25 deaths from Mountain Lions in North America since we began recording these in the 1800s.
By comparison, fatal shark attacks are over 6 times more likely to occur with about 1 shark attack in North America per year on average.
So it is interesting that whenever we go hiking in the woods or see a mountain lion cross the street, the news reports it and everyone loses their minds. Dying to a mountain lion is one of the rarest ways to die from a statistical perspective. There are rare diseases that you have never heard of that are more likely to kill you.
Every time we put our toe in the water people fear shark attacks. A rogue swell is more likely to capsize your boat than for you to be attacked by a shark, yet some people live in fear of it.
Also if you have hiked any considerable amount in the Rocky Mountains or Sierra Nevada mountain ranges than you have almost certainly had a Mountain Lion watch you, probably several times. Despite the fact, you have never seen it.
Bears are another one. I have encountered so many bears in the wild that I seriously don't even have a count for it anymore. I have stumbled on them less than 100 feet away from me on several occasions and guess what happens... they look me square in the eyes and then calmly turn around and walk away. Sometimes they run away, but they have no interest in killing me. I have never had one approach me once we became aware of the presence of each other.
I hike and camp A LOT, and I have NEVER seen a mountain lion in the wild, despite living in areas of their prime habitat where we know they live. I have seen virtually every other native large animal species on multiple occasions. But I have never seen a mountain lion in the wild. But I can guarantee you that they have seen me several times. That is what is scary about mountain lions. If you ever did get attacked, it would be over before you knew what was happening.
I have seen mountain lions cross the road twice in Park City, UT now. I have seen them in zoos as well. But just never seen one in the wild. Despite spending enough time to encounter bears on 20+ occasions.